Linear Regression for Single Prediction

Linear regression is a statistical method and machine learning foundation used to model relationship between a dependent variable and one or more independent variables. The primary goal is to predict the value of the dependent variable based on the values of the independent variables.

Predicting a Single Value Using Linear Regression

Once the model has been trained and evaluated we can use it to make predictions. When we talk about producing a single prediction value it means using a specific set of independent variable(s) to generate one dependent variable using linear regression model.

Example 1: Single Prediction Using Simple Linear Regression

Let’s assume you have a dataset that tracks hours studied (independent variable) and test scores (dependent variable). After training the model you want to predict the test score for a student who studied for 5 hours.

Using the equation of the regression line y = \beta_0 + \beta_1x, where: x is the number of hours studied.

We can substitute x=5x into the equation and compute y, which is the predicted test score.

Example 2: Single Prediction Using Multiple Linear Regression

In the case of multiple linear regression where more than one independent variable influences the dependent variable predicting a single value involves inputting multiple independent variable values in the model.

For example in predicting house prices factors such as area, number of bedrooms and location all be considered. Once you input the values for these independent variables the model will output predicted house price.

Building a Linear Regression Model

1. Loading and Preparing Data

First import necessary libraries and load the dataset. For this example we will generate a synthetic dataset with multiple features using numpy. Assume we are working with a dataset where we want to predict a target based on two random features.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
# Set random seed for reproducibility
np.random.seed(42)

# Generate random independent variables
n_samples = 1000
X1 = np.random.uniform(1, 100, n_samples)  # Random numbers for feature 1
X2 = np.random.uniform(1, 50, n_samples)   # Random numbers for feature 2

# Create a relationship between features and the target variable with some noise
noise = np.random.normal(0, 10, n_samples)
y = 2.5 * X1 + 3.8 * X2 + noise  # Linear relationship with noise

# Create a DataFrame for easy manipulation
data = pd.DataFrame({'Feature1': X1, 'Feature2': X2, 'Target': y})
data.head()

Output:

	Feature1	Feature2	Target
0	38.079472	10.071514	124.690605
1	95.120716	27.553146	334.234944
2	73.467400	43.774346	347.746226
3	60.267190	36.879019	294.481904
4	16.445845	40.521496	204.232146

This synthetic dataset has two features (Feature1 and Feature2) and one target variable (Target) with a linear relationship with some noise.

2. Exploratory Data Analysis (EDA)

Before moving to modeling let's analyze the data visually to ensure there is a linear relationship between the features and the target variable.

# Pairplot to visualize relationships between variables
sns.pairplot(data)
plt.show()

# Check the correlation between features and target
corr_matrix = data.corr()
sns.heatmap(corr_matrix, annot=True, cmap='coolwarm')
plt.show()

Output:

3. Data Preprocessing

Linear regression is sensitive to feature scaling. To improve model performance, it’s important to scale the features. We'll use StandardScaler from sklearn.

# Split data into independent variables (X) and dependent variable (y)
X = data[['Feature1', 'Feature2']]
y = data['Target']

# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Scale the features using StandardScaler
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

4. Train the Linear Regression Model

Now that the data is scaled, we can train the linear regression model. This step provides the model's coefficients and the intercept.

# Instantiate the linear regression model
model = LinearRegression()

# Train the model on the training data
model.fit(X_train_scaled, y_train)

# Check the model's coefficients and intercept
print("Coefficients: ", model.coef_)
print("Intercept: ", model.intercept_)

Output:

Coefficients:  [72.55199542 54.26405783]
Intercept:  224.59535042865838

5. Make Single Predictions

We can now make predictions on the test data and evaluate the model. Below, new_data represents a new data point with two feature values. The model will predict the target value based on this new input.

# Predict the target values for the test set
y_pred = model.predict(X_test_scaled)

# Example of predicting a single value using a new data point
new_data = np.array([[45, 30]])  # Example values for Feature1 and Feature2
new_data_scaled = scaler.transform(new_data)  # Scale the new data
single_prediction = model.predict(new_data_scaled)

print(f"Predicted value for the new data point {new_data[0]}: {single_prediction[0]}")

Output:

Predicted value for the new data point [45 30]: 226.37747796176552

6. Model Evaluation

To evaluate the model's performance we’ll compute the Mean Squared Error (MSE) and R-squared value for the test data.

# Calculate Mean Squared Error (MSE) and R-squared (R^2)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)

print(f"Mean Squared Error: {mse}")
print(f"R-squared: {r2}")

Output:

Mean Squared Error: 100.35222719050975
R-squared: 0.9876309086737035

Limitations of Linear Regression for Single Predictions

• Sensitivity to Outliers: Outliers have negative impact on linear regression models when making predictions. It can skew the best-fit line and result in inaccurate predictions.
• Overfitting: If the model is overly complex (for example, too many independent variable) it may fit the training data too closely leading to overfitting and perform poorly on new unseen data.
• Assumes a Linear Relationship: Linear regression assumes a linear relationship between the independent and dependent variables. However in real-world data is non-linear relationship.

Linear regression is a simple model that can be used to predict single values based on historical data. The process of building a linear regression model involves preparing data, training the model and making predictions based on specific inputs.